The Intensional Side of Algebraic-Topological Representation Theorems
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A First Introduction to the Algebra of Sentences
A First Introduction to the Algebra of Sentences Silvio Ghilardi Dipartimento di Scienze dell'Informazione, Universit`adegli Studi di Milano September 2000 These notes cover the content of a basic course in propositional alge- braic logic given by the author at the italian School of Logic held in Cesena, September 18-23, 2000. They are addressed to people having few background in Symbolic Logic and they are mostly intended to develop algebraic methods for establishing basic metamathematical results (like representation theory and completeness, ¯nite model property, disjunction property, Beth de¯n- ability and Craig interpolation). For the sake of simplicity, only the case of propositional intuitionistic logic is covered, although the methods we are explaining apply to other logics (e.g. modal) with minor modi¯cations. The purity and the conceptual clarity of such methods is our main concern; for these reasons we shall feel free to apply basic mathematical tools from cate- gory theory. The choice of the material and the presentation style for a basic course are always motivated by the occasion and by the lecturer's taste (that's the only reason why they might be stimulating) and these notes follow such a rule. They are expecially intended to provide an introduction to category theoretic methods, by applying them to topics suggested by propositional logic. Unfortunately, time was not su±cient to the author to include all material he planned to include; for this reason, important and interesting develope- ments are mentioned in the ¯nal Section of these notes, where the reader can however only ¯nd suggestions for further readings. -
The Gödel and the Splitting Translations
The Godel¨ and the Splitting Translations David Pearce1 Universidad Politecnica´ de Madrid, Spain, [email protected] Abstract. When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules be- came the dominant view until well into the new century. Recently, modal inter- pretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowl- edge bases. In this talk I explain how familiar embeddings of stable models into modal log- ics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel¨ in 1933 to em- bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that al- lows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an ade- quate provability interpretation of H within the Goedel-Loeb logic GL, the sys- tem shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. -
Modal Logics That Bound the Circumference of Transitive Frames
Modal Logics that Bound the Circumference of Transitive Frames. Robert Goldblatt∗ Abstract For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case n =0 is the G¨odel-L¨ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When n = 1, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to in- tuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily n +1-irresolvable spaces when the modality ♦ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of ♦ as the derived set (of limit points) operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreovereach algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them. 1 Algebraic Logic and Logical Algebra The field of algebraic logic has been described as having two main aspects (see the introductions to Daigneault 1974 and Andr´eka, N´emeti, and Sain 2001). -
Basic Category Theory and Topos Theory
Basic Category Theory and Topos Theory Jaap van Oosten Jaap van Oosten Department of Mathematics Utrecht University The Netherlands Revised, February 2016 Contents 1 Categories and Functors 1 1.1 Definitions and examples . 1 1.2 Some special objects and arrows . 5 2 Natural transformations 8 2.1 The Yoneda lemma . 8 2.2 Examples of natural transformations . 11 2.3 Equivalence of categories; an example . 13 3 (Co)cones and (Co)limits 16 3.1 Limits . 16 3.2 Limits by products and equalizers . 23 3.3 Complete Categories . 24 3.4 Colimits . 25 4 A little piece of categorical logic 28 4.1 Regular categories and subobjects . 28 4.2 The logic of regular categories . 34 4.3 The language L(C) and theory T (C) associated to a regular cat- egory C ................................ 39 4.4 The category C(T ) associated to a theory T : Completeness Theorem 41 4.5 Example of a regular category . 44 5 Adjunctions 47 5.1 Adjoint functors . 47 5.2 Expressing (co)completeness by existence of adjoints; preserva- tion of (co)limits by adjoint functors . 52 6 Monads and Algebras 56 6.1 Algebras for a monad . 57 6.2 T -Algebras at least as complete as D . 61 6.3 The Kleisli category of a monad . 62 7 Cartesian closed categories and the λ-calculus 64 7.1 Cartesian closed categories (ccc's); examples and basic facts . 64 7.2 Typed λ-calculus and cartesian closed categories . 68 7.3 Representation of primitive recursive functions in ccc's with nat- ural numbers object . -
Distribution Algebras and Duality
Advances in Mathematics 156, 133155 (2000) doi:10.1006Âaima.2000.1947, available online at http:ÂÂwww.idealibrary.com on Distribution Algebras and Duality Marta Bunge Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 Jonathon Funk Department of Mathematics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6 Mamuka Jibladze CORE Metadata, citation and similar papers at core.ac.uk Provided by ElsevierDepartement - Publisher Connector de Mathematique , Louvain-la-Neuve, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium; and Institute of Mathematics, Georgian Academy of Sciences, M. Alexidze Street 1, Tbilisi 380093, Republic of Georgia and Thomas Streicher Fachbereich 4 Mathematik, TU Darmstadt, Schlo;gartenstrasse 7, 64289 Darmstadt, Germany Communicated by Ross Street Received May 20, 2000; accepted May 20, 2000 0. INTRODUCTION Since being introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [59, 12, 15, 19, 24]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) lattice-theoretic notion of dis- tribution algebra. We characterize the distribution algebras in E relative to S as the S-bicomplete S-atomic Heyting algebras in E. As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [7, 10, 12]. 133 0001-8708Â00 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. -
On the Blok-Esakia Theorem
On the Blok-Esakia Theorem Frank Wolter and Michael Zakharyaschev Abstract We discuss the celebrated Blok-Esakia theorem on the isomorphism be- tween the lattices of extensions of intuitionistic propositional logic and the Grze- gorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimplication. In memory of Leo Esakia 1 Introduction The Blok-Esakia theorem, which was proved independently by the Dutch logician Wim Blok [6] and the Georgian logician Leo Esakia [13] in 1976, is a jewel of non- classical mathematical logic. It can be regarded as a culmination of a long sequence of results, which started in the 1920–30s with attempts to understand the logical as- pects of Brouwer’s intuitionism by means of classical modal logic and involved such big names in mathematics and logic as K. Godel,¨ A.N. Kolmogorov, P.S. Novikov and A. Tarski. Arguably, it was this direction of research that attracted mathemat- ical logicians to modal logic rather than the philosophical analysis of modalities by Lewis and Langford [43]. Moreover, it contributed to establishing close connec- tions between logic, algebra and topology. (It may be of interest to note that Blok and Esakia were rather an algebraist and, respectively, a topologist who applied their results in logic.) Frank Wolter Department of Computer Science, University of Liverpool, U.K. e-mail: wolter@liverpool. ac.uk Michael Zakharyaschev Department of Computer Science and Information Systems, Birkbeck, University of London, U.K. -
Topos Theory
Topos Theory Olivia Caramello Introduction Interpreting logic in categories First-order logic First-order languages First-order theories Categorical semantics Topos Theory Classes of ‘logical’ categories The interpretation of logic in categories The interpretation of formulae Examples Soundness and completeness Olivia Caramello Toposes as mathematical universes The internal language Kripke-Joyal semantics For further reading Topos Theory Interpreting first-order logic in categories Olivia Caramello Introduction • In Logic, first-order languages are a wide class of formal Interpreting logic in categories languages used for talking about mathematical structures of First-order logic any kind (where the restriction ‘first-order’ means that First-order languages quantification is allowed only over individuals rather than over First-order theories collections of individuals or higher-order constructions on Categorical semantics them). Classes of ‘logical’ categories • A first-order language contains sorts, which are meant to The interpretation of formulae represent different kinds of individuals, terms, which denote Examples individuals, and formulae, which make assertions about the Soundness and completeness individuals. Compound terms and formulae are formed by Toposes as mathematical using various logical operators. universes • It is well-known that first-order languages can always be The internal language interpreted in the context of (a given model of) set theory. In Kripke-Joyal semantics this lecture, we will show that these languages can also be For further reading meaningfully interpreted in a category, provided that the latter possesses enough categorical structure to allow the interpretation of the given fragment of logic. In fact, sorts will be interpreted as objects, terms as arrows and formulae as subobjects, in a way that respects the logical structure of compound expressions. -
Esakia's Theorem in the Monadic Setting
Esakia’s theorem in the monadic setting Luca Carai joint work with Guram Bezhanishvili BLAST 2021 June 13, 2021 • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) • Translation of intuitionistic logic into modal logic (G¨odel) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic -
Constructive Embeddings of Intermediate Logics
Constructive embeddings of intermediate logics Sara Negri University of Helsinki Symposium on Constructivity and Computability 9-10 June 2011, Uppsala The G¨odel-Tarski-McKinsey embedding of intuitionistic logic into S4 ∗ 1. G¨odel(1933): `Int A ! `S4 A soundness ∗ 2. McKinsey & Tarski (1948): `S4 A ! `Int A faithfulness ∗ 3. Dummett & Lemmon (1959): `Int+Ax A iff `S4+Ax∗ A A modal logic M is a modal companion of a superintuitionistic ∗ logic L if `L A iff `M A . So S4 is a modal companion of Int, S4+Ax∗ is a modal companion of Int+Ax. 2. and 3. are proved semantically. We look closer at McKinsey & Tarski (1948): Proof by McKinsey & Tarski (1948) uses: I 1. Completeness of intuitionistic logic wrt Heyting algebras (Brouwerian algebras) and of S4 wrt topological Boolean algebras (closure algebras) I 2. Representation of Heyting algebras as the opens of topological Boolean algebras. I 3. The proof is indirect because of 1. and non-constructive because of 2. (Uses Stone representation of distributive lattices, in particular Zorn's lemma) The result was generalized to intermediate logics by Dummett and Lemmon (1959). No syntactic proof of faithfulness in the literature except the complex proof of the embedding of Int into S4 in Troelstra & Schwichtenberg (1996). Goal: give a direct, constructive, syntactic, and uniform proof. Background on labelled sequent systems Method for formulating systems of contraction-free sequent calculus for basic modal logic and its extensions and for systems of non-classical logics in Negri (2005). General ideas for basic modal logic K and other systems of normal modal logics. -
Induced Morphisms Between Heyting-Valued Models
Induced morphisms between Heyting-valued models Jos´eGoudet Alvim∗ Universidade de S˜ao Paulo, S˜ao Paulo, Brasil Arthur Francisco Schwerz Cahali† Universidade de S˜ao Paulo, S˜ao Paulo, Brasil Hugo Luiz Mariano‡ Universidade de S˜ao Paulo, S˜ao Paulo, Brasil December 4, 2019 Abstract To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that de- termine them: the only cases found in the literature are concerning automorphisms of complete Boolean algebras and complete embedding between them (i.e., injec- tive Boolean algebra homomorphisms that preserves arbitrary suprema and arbi- trary infima). In the present work, we consider and explore how more general kinds of morphisms between complete Heyting algebras H and H′ induce arrows ′ between V(H) and V(H ), and between their corresponding localic toposes Set(H) ′ (≃ Sh (H)) and Set(H ) (≃ Sh (H′)). In more details: any geometric morphism ′ f ∗ : Set(H) → Set(H ), (that automatically came from a unique locale morphism ′ f : H → H′), can be “lifted” to an arrow f˜ : V(H) → V(H ). We also provide also ′ some semantic preservation results concerning this arrow f˜ : V(H) → V(H ). Introduction The expression “Heyting-valued model of set theory” has two (related) meanings, both arXiv:1910.08193v2 [math.CT] 3 Dec 2019 parametrized by a complete Heyting algebra H: (i) The canonical Heyting-valued models in set theory, V(H), as introduced in the setting of complete boolean algebras in the 1960s by D. Scott, P. -
Hilbert Algebras As Implicative Partial Semilattices
DOI: 10.2478/s11533-007-0008-2 Research article CEJM 5(2) 2007 264–279 Hilbert algebras as implicative partial semilattices J¯anis C¯ırulis∗ University of Latvia, LV-1586 Riga, Latvia Received 15 August 2006; accepted 16 February 2007 Abstract: The infimum of elements a and b of a Hilbert algebra are said to be the compatible meet of a and b, if the elements a and b are compatible in a certain strict sense. The subject of the paper will be Hilbert algebras equipped with the compatible meet operation, which normally is partial. A partial lower semilattice is shown to be a reduct of such an expanded Hilbert algebra iff both algebras have the same filters. An expanded Hilbert algebra is actually an implicative partial semilattice (i.e., a relative subalgebra of an implicative semilattice), and conversely. The implication in an implicative partial semilattice is characterised in terms of filters of the underlying partial semilattice. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Compatible elements, filter, Hilbert algebra, implication, implicative semilattice, meet, partial semilattice MSC (2000): 06A12, 03G25 1 Introduction Hilbert algebras, or positive implication algebras [35], are the duals of Henkin algebras called by him implicative models in [21] (this term has also been used for Hilbert algebras [30, 31]). Positive implicative BCK-algebras [24] are actually another version of Henkin algebras (see, e.g. [28, 37]). As a matter of fact, these algebras are an algebraic counterpart of positive implica- tional calculus. Various expansions of Hilbert algebras by a conjunction-like operation have also been studied in the literature. -
Some Glances at Topos Theory
Some glances at topos theory Francis Borceux Como, 2018 2 Francis Borceux [email protected] Contents 1 Localic toposes 7 1.1 Sheaves on a topological space . 7 1.2 Sheaves on a locale . 10 1.3 Localic toposes . 12 1.4 An application to ring theory . 13 2 Grothendieck toposes 17 2.1 Sheaves on a site . 17 2.2 The associated sheaf functor . 19 2.3 Limits and colimits in Grothendieck toposes . 21 2.4 Closure operator and subobject classifier . 22 3 Elementary toposes 25 3.1 The axioms for a topos . 25 3.2 Some set theoretical notions in a topos . 26 3.3 The slice toposes . 28 3.4 Exactness properties . 29 3.5 Heyting algebras in a topos . 30 4 Internal logic of a topos 33 4.1 The language of a topos . 33 4.2 Interpretation of terms and formulæ . 35 4.3 Propositional calculus in a topos . 37 4.4 Predicate calculus in a topos . 38 4.5 Structure of a topos in its internal language . 40 4.6 Boolean toposes . 41 4.7 The axiom of choice . 42 4.8 The axiom of infinity . 43 5 Morphisms of toposes 45 5.1 Logical morphisms . 45 5.2 Geometric morphisms . 46 5.3 Coherent and geometric formulæ . 48 5.4 Grothendieck topologies revisited . 49 5.5 Internal topologies and sheaves . 50 5.6 Back to Boolean toposes . 52 3 4 CONTENTS 6 Classifying toposes 53 6.1 What is a classifying topos? . 53 6.2 The theory classified by a topos . 54 6.3 Coherent and geometric theories .